THE MCHHMMAL URNAL T(OFM Y(M. aNO. SPRN99 \ EL43 VI . S

The 20-knot Human-Powered Water Craft

(HP is reprinting, in parts,Alec Brooks'paperon high-speed Introduction hydrofoils, which was given at the Third IHPVA Scientific Symposium. It has been in great demand. As a result of this For the past hundred years, the highest-speed human- paper, the article in the SCIENTIFIC AMERICAN in Decem- powered water craft have been the narrow oared racing ber 1986, and demonstrationsby and interviews of Allan Ab- shells. Under the sanctioning of collegiate bott and Alec Brooks on TV and other media, including an ap- organizations, these craft evolved slowly into the elegant pearance on Australian television in a programcalled "Beyond craft of today. This evolution has much in common with that of the standard racing . Both are highly optimized 2000", individuals, groups and universities around the world are vehicles which evolved under a set of constraints designed to working feverishly on human-powered hydrofoils. I asked force all vehicles of each class to be largely the same. Under Alec on April 16 what developments there had been since the these contraints progress is made, but only in gradual Vancouver EXPO races and symposium in September 1986, refinements, never in great leaps. and he said that Allan and he had been doing a little develop- ment by testing their two Flying Fishes against each other; The IHPVA was founded to foster innovation in the that two commercial companies were discussing possible pro- design of human-powered vehicles. As a result of the IHPVA, land-vehicle performance has taken a quantum leap duction; and that they were preparingfor a June assault on the in the last decade. Freed of restrictive design rules, designers 2000-m record with Steve Hegg providing the power. -- Dave have developed streamlined land vehicles which are 50 Wilson) percent faster than standard . A similar revolution is just beginning in human-powered water vehicles. With the encouragement and sanctioning of the IHPVA, a new wave of Abstract watercraft will soon be toppling all existing records held by oared shells, with speed improvements at least as great as Until now, the fastest human-powered water craft have those seen for the land vehicles. continued on page 8 been the long narrow oared shells, powered by crews of between one and rowers. Recent developments in hydrofoil and propeller design have resulted in several In this issue... single-rider water craft that are as fast, or faster, than single shells. These craft are in the infancy of their development. With further refinement, single-rider hydrofoils will be The 20 Knot Human Powered faster than oared eights. A method is given for optimum Water Craft, Alec Brooks 1 wing sizing, based on structural factors, minimum power, Editorial Comments, and takeoff speed constraints. The main design parameter is Dave Wilson 2 the expected duration of the competition -- this determines Letters to 3 how much power is available from the rider. Other design Conception and Optimization of considerations are the takeoff speed and minimum power Human Powered Aircraft, required. For very short, high-power cases, the optimum E. Schoberl 4- 14 design has very small wings and a high takeoff speed, making it impossible to ride at low speeds or low power. High power Dynamic Stability of the Bicycle, designs with constraints on minimum power and takeoff Y.H6naff 15 -17 speed are explored, showing the associated penalties in top Point/Counterpoint, speed. It is shown that below a certain design power level, H6naff, Papadopoulos & hydrofoils, while still possible, are less efficient than Ruina 18 -19 displacement-hulled craft. Build the Flunder, Reiss and Pivit 20 I tpage 2 oae H1JMANPnWFR.UMN PWR.Sri~ Snrinq,. 181987 vol6no1vol 6 no. I EDITORIAL COMMENTS by David Gordon Wilson HUMAN POWER The Technical Journal of the This issue has been the easiest to put together in the three years of my editorship, mainly INTERNATIONAL HUMAN- because of the very great help of the volunteers named on this page. There has also been another POWERED VEHICLE significant development: contributions have been sent to HUMAN POWER unsolicited. And they ASSOCIATION are very good contributions, as I hope you will agree. It is a compliment. The authors feel that David Gordon Wilson, editor HP is a worthy publication to carry the fruits of their considerable labors. May the trend continue! 15 Kennedy Road All we need now to advance to the point where we can further improve the scope, size and CAMBRIDGE, MA 02138, quality of our journal is a readership in the ten-to-fifty-thousand range. Then we could ease the load USA on our volunteers and bring in some commercial-advertising income. The IHPVA has been Phones: 617-253-5121 (work) getting many excellent notices in prestigious publications. We need to translate this enthusiasm 617-876-6326 (home) into IHPVA members. Having the international championships as part of "HUMAN-POWER Spring 1987 volume 6, no. 1 WEEK" in Washington, D.C. in September, as Marti Daily, Steve Stollman and others are working hard to bring to reality, could be the key to our growth.

IIUMAN POWER is UNRESTRAINED BRACHISTOCHRONES published quarterly by the International Human Powered You may feel that the last diversion you would want in an attempt on the Vehicle Association, Inc., a would be an unrestrained ant, let alone a brachistochrone. You may be wrong. I met the world's non-profit organization devoted expert on these recently at Clark University: David G. Stork, of the physics faculty. He sent to the study and application of me several questions of the following type. Suppose that two HPVs are travelling on a human muscular potential to frictionless horizontal plane with the same initial velocity. At point A one continues on the plane propel craft through the air, in and arrives at point B. The other takes a route that passes into an undulating valley, ending up the water and on land. with a climb to point B. Which is going faster at point B? Answer: They are moving at the same Membership information is speed, because there is no mechanism to introduce friction. Which gets there first? Answer: The available by sending a self- HPV that went through the valley will always get there first, even though it traverses a longer addressed, stamped business- distance. Why? Because its average velocity along the slope, and its horizontal component of sized envelope to: IHPVA velocity, are higher than that of the HPV on the plane. Therefore it is possible to have an P.O. Box 51255, undulating track with a mean slope within IHPVA limits that will have local deviations from the Indianapolis, IN slope and that will give possibly considerable advantages over a track with a constant slope. 46251-0255 USA. David Stork has promised us a summary article on sliding and rolling brachistochrones for a future issue. Material in HUMAN POWER A HISTORY OF THE IHPVA SPEED CHAMPIONSHIPS is copyrighted by the IHPVA. Unless copyrighted by the Ronald Steven Blair is writing a history of the championships, compiling the rules as they author(s), complete articles or evolved and the currently valid rules, and will later be making the history available for purchase. representative excerpts may be This is the type of individual enterprise that makes the IHPVA what it is. If anyone has material published elsewhere if full to contribute, please let Steve know: 275 Castle Hill Ranch Road, Walnut Creek, CA history of credit to the author(s) and the 94595; phone 415-932-6460. Some time I plan to make an index and a one-page IHPVA is prominently given. HUMAN POWER. If anyone has made a start, would s/he please let me know.

This issue of HUMAN THE FARNSWORTH FORMULA POWER was typed by Sabina believes (and I Ratj, Dan Bloom and David Jim Farnsworth is a man with a mission: to change the bicycling world. He Wilson; was set up by Tom agree with him) that it makes no sense to have a standard crank length for almost all sizes of Healy and Marti Daily. Printed bicycles. Jim would go further than that: he believes that the crank length should approach half the major cause of knee injuries among cyclists." at Apple Press by Tom Healy. length of the femur, "The 170-mm crank may be the long. He also believes - as I Distributed by Marti Daily and He is of about average height, and uses cranks a little over 220 mm achieved through a large- a cast of dozens. Special do - in wide-range gears. He believes that optimum foot speed is better at high rpm. One reason is thanks to translators, including diameter pedal circle at moderate rpm than with a small-diameter circle Incidentally, Otto Brodtrick, who translated that the dynamic forces, for the same foot speed, go down with increase of radius. on a "regular" ten-speed bike. the "Flunder" article. long cranks require less of a compromise on a recumbent than Jim Farnsworth lent me the manuscript of an entertainingly written book about his adventures in design and in life. If a publisher were to want to look at it, he would be delighted. Here is his address: 398 Shore Drive, Laconia NH 03246 USA 603-524-1695.

- HUMAN POWER, Spring, 1987 vol. 6 no. 1 na P 3 AX LETTERS is a complex matter. We have much to the ground on December, 6, 1985. This IN THE AIR learn from around the world if we are to was the ninth day of experimentation and create a technically advanced and humane the 85th trial flight. The pilot was his stu- The free exchange of ideas is important HPUV usage. dent Koichi Nakamura, Japanese champion to R & D, and HUMAN POWER does this There are a few points that Fred Willkie long-distance hang-glider pilot. Unfortu- very well... [I wanted to send a report on has mentioned that I feel are important for nately, there was no film or tape record of a] book on unconventional aircraft... any locale. the flight and he feels that he cannot claim compact circular wings or circular wing the flight without corroboration. types shown in the book might be practi- 1. Materials. 2. After his retirement from Nihon Uni- cal for recreational HPAs. It is Unconven- I am as interested as the next person in versity, there is no successor wanting to tional Aircraft, by Peter M. Bowers, Tab lightening these vehicles and making them carry on his work either at Nihon or else- Books, Blue Ridge Summit, PA 17214, more efficient. However, it is easy to feel where in Japan. 1984. [Here is a list of topics:] creatively impotent these days if one isn't Platz sailboat glider; Zanonia seed-type using the latest high-tech materials. Use Toshio Kataoka 914-6 Mamedo Ko- aircraft; advantages of tail-less aircraft for available materials! Many Europeans I huku, Yokomama Kanagawa, Japan hang gliders and ultralights in the 1970s contacted utilize recycled materials in their 222 and 1980s; circle-wing biplane; American construction of HPUVs. A few pieces of Nemuth parasol, cirular wing; Arup flying tubing and a wheel - voila - a sidecar! And (And in a later letter Toshio Kataoka wings; Flettner-rotor wing; Lanier Vacu- it isn't likely that you will have to melt sent a copy of a report of the 12th IIPSC plane; Custer channel wing; landing gear; down old buckles for brazing wire! in the leading newspaper Yomiuri Shinbon ducted propellers; Ben Brown's diamond of January 22, 1987, and wrote wing. 2. The Social and Economic Context "Congratulations! of HPA of Unlike Bangladesh, where the motiva- MIT!" He enclosed a couple of color prints, Edward G. Sward, tion to use these vehicles seems to be pri- which we may be able to reproduce here -- 215 Cambridge St., Worcester, marily related to economics, we in Western Dave Wilson.) MA 01603 USA cultures must emphasis other motivations in advocating HPUVs. So, in the words of STUDDED TIRES Fred Willkie "... have the right questions before you offer answers." For many peo- COMMENTS ON "" Here is an update on the studded bike ple will want to know what place these cra- tires. On page 14 of CYCLIST, April, zy vehicles could have in their lives. The first thought that came to my mind 1987, there are photographs of a set of Whether it be providing substitute trans- after rushing through the piece, tires with #6 pan-head sheet-metal screws port methods for the most polluting of ur- 'Rickshaws in Bangladesh' by Fred Will- and nuts. I have since tried these screws, ban vehicles, the van and taxi; or for kie, was that he was saying a lot about plus washers under the nuts, in my tires. transporting children and handicapped, or everything, and doing it very well. As an They are better than those reported in my for cleaning streets, or ..... "the fact is that advocate for HPUV usage, specifically in article in HP 5/4 because the screws are one must consider what will work in the non-third-world locales, I felt that his hardened, and the points are sharp for bet- context of people's minds and lives. De- stark accounts of the social and economic ter traction on ice. I did a quick experiment velopment without context is like a bicy- realities of drivers were relevant by grinding two screws side by side. The cle tire in a tram track. Very linear! to HPUV development in Western coun- hardened screw took almost twice as much Thanks Fred Willkie. tries as well. (Recently, I heard of finan- grinding to remove the same amount of cially abused delivery drivers in New York metal. The sheet-metal screws tap their Jan Vandertuin City in the Asian-restaurant business). own threads in the nuts, making them a P.O. Box 573 My own intercultural experiences with little more difficult to get started, but the South Edgemont, MA 01258 USA HPUV development began in a slightly finished result gives superior traction and different culture. Yet somehow, the simi- greater durability. larities between his conclusions and my HP HELICOPTERS own are not far apart. James Donohue, 87 Plymouth While researching and eventually con- (This was in reply to my letter enquiring Drive North, Glen Head, NY 11545 structing trailers for the distribution of about the Nihon University work on HP USA food in Switzerland, I often had a vision helicopters Ed.) of a culture where this mode ... I am only an HPV fan. I am not a FIGURE-EIGHT DRIVE would be developed. It wasn't just a vi- member of the Nihon University HP Heli- sion of advances in HPUV technical devel- copter (HPH) team. But I did visit Professor Thank you for publishing my article opment but included improvements in life, Naito at his home, and told him about the about my figure-eight drive in IIUMAN twelfth IHPSC. I asked him if he could re- specifically in how we move goods and POWER: it was an honor to present my port on the latest developments on the hel- people. discovery to our HP group. It seemed as if Asia was the place where icopter. He declined for the following rea- sons. HPUVs had become established. Yet after Anthony J. Patroni, 9005 Amherst 1. He succeeded in hovering his HPH for reading of the situation in Bangladesh, it Avenue, Margate NJ 08402 USA a few seconds several centimeters above is obvious that a HPUV transport system HUMAN POWER, Srini, 1987 vol n. page 4 ·J 11! · ~ ~ ~ ~ ~ ~ CONCEPTION AND OPTIMIZATION OF HUMAN- IAIF"r''" " r% A I r'h`U'A T" Pfft UV r r, Il i LA I W t -W 16& r--' I i--- I I lw WV 1 g % lI * I

By E. Schoberl

(This is an edited and shortened version of a paper presented at the twentieth OSTIV-Congress at Benalla, Australia in January, 1987. (It is both a summary of

and a sequel to the article on the WING SPAN 2 WING AREA 18.2m 2 by the same author ASPECT RATIO 34.3 MUSCULAIR1 and AIRFOIL FX 76 MP EMPTY WEIGHT 31 kg in HUMAN POWER vol. 5, no. 2, GROSS WEIGHT 85 kg OPTIMUM 8.5m/s summer 1986. - Dave Wilson) FLYING SPEED MINIMUM POWER 190 WATT < REQUIREMENT MAX. GLIDE RATIO 44 Summary lm 5m The progress made in fibre-reinforced I - composites and the resulting advanced aer- FIG. 1 The cantilevered high-wing monoplane Cyclair combines minimum power demand odynamical design (laminar-flow airfoils) with good-natured flying behavior. over the past 15 years has made it possi- - sandwich-covered cabin with semi- crossed the English Channel in 1979 ble to build human powered planes with recumbent pilot position with the GOSSAMER ALBATROSS, half the previous weight, and of aerody- - all-moving elevator setting two incomparable milestones in namically sophisticated design. - all-moving rudder with auxiliary the history of . Conventional high - wing monoplanes rudder The Kremer World Speed Competition with faired hanging cabin and all - moving - rear propeller to protect the plane demanded a flight over a 1500-m-long rudders have been tested with great suc- from avoidable turbulences triangular course to be covered within cess. The German planes MUSCULAIR I three minutes and allowed storage of the and II are to be considered as aerodynami- This CYCLAIR plane is similar to pilot's energy ten minutes prior take off. cally highly sophisticated as they have no the MIT- designed MICHELOB LIGHT This prize was first won by an MIT stu- wire bracing and no energy storage. With EAGLE which is larger, a little heavier dent group with their MONARCH in MUSCULAIR II an average athlete im- and optimized for slower flying speed. May 1984. Four further prizes have proved on the record speed in the Kremer With this MICHELOB LIGHT EAGLE, been awarded for flights to improve the World Speed Competition with about half MIT will make the flight legend of Dae- existing record by at least five percent. the power requirement compared to the dalus a reality in the near future. MacCready did this with the Bionic Bat rest of the competitors. This 110-km-long flight from Crete in July and December 1984. Both MIT The whole range of possible flying to the Greek mainland will be the cli- and MacCready used energy storage. speeds were investigated for human- max and perhaps the end of the initial Also in 1984, a German amateur powered planes of optimum design at the phase of human-powered flight which team, unsupported by any large wealthy present state of technology. A minimum will remain a privilege for few enthu- sponsor, built their first human-powered power requirement of under 200 watts is siasts only because it requires both a aircraft, MUSCULAIR 1. Two weeks possible with extremely large planes ,,high-grade technology and a high-level after completing the MUSCULAIR 1 which are very difficult to handle and to of effort of the pilot. they won the figure-eight Kremer prize fly. The prizes donated by Henry Kremer for non-Americans with a speed almost There is an optimum configuration at were worldwide a great incentive for the twice as fast as MacCready's GOSSAM- a flying speed of about 8.5 m/s. This development of many human-powered ER CONDOR. Two months later they plane with about 25 m wing span has a aircraft. The precise design of high- won the Kremer speed prize in August, power requirement slightly below 200 strength ultralight construction is also 1984 by improving MacCready's speed watts (near the ground) with a glide ratio necessary for planes with the lowest of the BIONIC BAT by seven percent, of 44 and can be flown by experienced possible power requirement like electric without energy storage. light-weight pilots. or solar-powered planes or unmanned To test the reserves of pilot and air- CYCLAIR can be built at a weight high-altitude crafts for communication craft they made the first human-powered of slightly over 30 kg with the following relays in the stratosphere. passenger flight in the history of avia- design characteristics. tion on October 1, 1984. - fully sandwich-covered wings with General In 1985 this airplane was involved laminar-flow airfoil Bryan Allen was the first who com- in a traffic accident on the road and was - carbon-fibre-reinforced epoxy main pleted the one mile long, figure-eight heavily damaged. The team decided to spar in four sections designed for three flight with Paul MacCready's GOS- build the MUSCULAIR 2, being an air- times the static load SAMER CONDOR in 1977, and craft designed for high speed. Again HUMAN POWER, Srine;, 1987 vol. 6 no. nae 5 H i 18 vo.D-o.1-]i-5 they won the Kremer speed prize and im- * the unavoidable side-faces of the Optimization from long- proved their world record from 35.7 kmin/ fuselage located far in front of the distance flight to high-speed h to 44.3 km/h. center of gravity make yaw control flight difficult. Sometimes a power- To get an impression of the size and consuming swept-back main-wing Based on the just-mentioned consider- weight of the airplanes: design is necessary; ations and on the aircraft weights that * the lift of the canard wing is can be reached with advanced aerody- MUSCULAIR1: span 22 m; generated with too much induced namic microlight design, the whole empty weight 28 kg drag because of its low aspect ratio; range of human-powered flight was in- MUSCULAIR 2: span 19.5 m; * the main wing often does not vestigated from long-distance flight empty weight 25 kg operate at the best glide ratio or with minimum power requirement to minimum sinking velocity, as its short-time higher-speed flights. This paper deals mainly with experi- angle of attack cannot be adjusted as Only the best laminar-flow airfoils in ences gained by the MUSCULAIR pro- can that of the canard wing; and the range of Reynolds numbers from jects and with possible developments * the canard wing affects the field of 300,000 to 800,000 were considered, corresponding to the present state of the vision. such as the FX 76 MP, designed in art as well. 1976 by Prof. F. X. Wortmann espe- Conventional high-wing monoplanes cially for HPAs. General considerations of with faired hanging cabins and all- The following main influences were plane conceptions moving rudders have proven to be the taken into consideration in making the best for light-weight, efficient, safe and optimizations: Although many pioneers achieved simple controllable HPAs. * aircraft dimensions and their great successes with canard-type With the increased knowledge of the corresponding weights; planes (Wright brothers, Focke, Mac- precise design of high-strength micro- * ground-effect factor of 0.8 (flying Cready and the latest, Burt Rutan and light construction, designers will turn altitude aboutl/4 wing span); Jeana Yeager with their Voyager in Dec. to cantilevered structures that eliminate * lift coefficient not over 1.1 in order 1986) this conception could, despite its external wires. Rough estimations at to have enough stability and control stall safety, not represent a break- the early design state of MUSCULAIR reserves; through. 1 clearly showed that the power de- * Reynolds numbers not below The following four disadvantages are manded due to the increase of structural 300,000 to avoid unexpected mainly responsible: weight is more than outweighed by the laminar-bubble separation with lower drag of a cantilevered design. The hysteresis effects; and contribution of ground effect from the * minimum chord length at the wing use of low-winged monoplanes was of a low-speed long-distance plane, generally overestimated. 0.7 m; a high-speed plane, 0.4 m Spring-neutralized all-moving eleva- tors and rudders turned out to be both Figure 2 shows the rapid decrease light-weight and effective. Based on the in wing area with increasing flying analyses of test flights the author speed. thinks that an all-moving rudder with The optimum wing span and chord an auxiliary rudder is more favorable. length of a trapezoid wing with the cor- d' Front, middle and rear propeller posi- responding sinking velocity and power tions have proven equivalent in success- requirement is given infigure 3 for the 9 ful HPAs, although the middle and rear- whole flying-speed range. arrangements are aerodynamically bet- The most important results of the ter. optimization are shown in the velocity A favorable aircraft concept com- polars (fig. 4). The minimum power bining minimum power demand and requirement of HPA at the present state good-natured flying behavior is the can- of the art is slightly more than 150 tilevered high-wing monoplane CY- watts (near the ground), achievable with CLAIR (Fig. ) with laminar-flow air- a low-speed plane (about 6 m/s) of over foils, hanging faired cabin with the pi- 25-m wing span. The author is of the lot in a semirecumbent position, all- opinion that wing-spans larger than 25 moving elevator, all-moving rudder m are not practical, because this gives with auxiliary rudder and pusher propel- nearly no reduction in power require- FLIING SPEED r./9] ler. ment, but the aircraft becomes hard to FIG. 2: Wing Area Required for The design considerations of this control because of the increased mo- CL = 1.1 new design will now be discussed. page 6 HUMAN POWER, Sprin, 1987 I Jl vol. 6 no. 1 eJ

er Reynolds numbers there. It was found that the pilot delivers _ _ _ -W MG RO T- IN T -. The increase in induced drag near the about the same power output in semi- _-: KS.._.. I b O-- .. ground is small when turning away from recumbent position as in the upright po- the '1 ideal elliptical lift distribution. More sition. In the semi-recumbent position ; important are well-shaped and profiled the cabin can be made smaller. The aero- 55 wing tips. -0 13 /s 2. - With the semicircular design dynamic drag can be further reduced of the wing tips the effective wing span through the use of an extremely light fi- is increased and hence the induced drag is berglass sandwich fairing at least in the slightly reduced. 6 0. 5 front part which ensures a more accurate IY r To build a light-weight, strong shape and surface finish. wing that is stiff torsionally and in bend- The supporting fuselage structure with 0 75 0. 2 ing with the necessary accuracy required the wing connections and the stabilizer I for laminar-flow airfoils, the application strut in which the propeller shaft rotates f 0; 67 of carbon-fibre-reinforced spar and foam/ are preferably made of carbon-fibre tubes. fiberglass sandwich covering is necessary OP IN IN (fig. 5). For aerodynamic reasons the Drive and propulsion . sandwich has to cover at least the whole laminar area on the upper side and from Although the flapping wing can in the- FIG.: 3, Min. sinking velocity and power the nose to the curvature inflection on ory be made quite efficient, the propeller requirement and optimum wing configura- the lower side. In order to get the wing is by far the most efficient means of pro- tion of man-powered planes. torsionally stiff enough the nose cover- pulsion. A maximum efficiency of 90 ment of inertia, even with extremely ing and the spar should form a closed percent can be reached in HPA only with lightweight construction. In addition, tube. However, the sandwich covering relatively large slowly running propellers aircraft become quite sensitive to gusts. and the carbon spar deform differently un- designed for minimum induced losses. A favorable wing has 25-m span der load. On the wings of the MUSCU- Larrabee's elegant method hasto be men- (18m2 wing area), has a maximum glide LAIR 2 boundary-layer transition oc- tioned here. ratio of 44, and a power requirement of curred too early just behind the spar on A large propeller adds weight and be- only 190 watts. Such a wing can be sat- the upper side. This can be avoided with yond a certain= size the tips are likely to isfactorily handled and controlled by an a slightly flexible skin covering the spar. strike the ground when the aircraft is tak- experienced pilot. The best solution seems to be a wing- ing off or landing. Therefore the diameter This CYCLAIR-type of plane is capa- structure as shown in Fig. 5 with a flexi- required for good efficiency (89 percent) ble of covering long-distance flights, ble aileron connection forming an inte- and medium efficiency (85 percent) is giv- even against light headwinds, and does gral part. These flexible connections en in fig. 6. not react so sensitively to gusts because have been tested successfully in gliders The chain drive is highly efficient, of the higher flying speed. (Speed Astir) and model aircraft. ,.ttwnnlmrmtmrmr rmln mmnanr mrmlmrmImtpr mf' vrp At a flight speed of more than 13 m/s 3' a very high power is required with an air- Elevator and rudder 0. .Lr plane with smaller wingspan. The high- All-moving elevators and rudders, ap- speed aircraft MUSCULAIR 2 showed IX that first-class athletes are capable of plied mainly for weight reasons, have z 45 .17 maintaining such speeds for a few min- generally proved to be good. An all- 0.1 utes only. With such an airplane it moving vertical tailplane with auxiliary seems possible to fly the 1500-m speed rudder seems to be better for lateral con- RA 7- course in about 100 seconds without en- trol. Airfoils with 9-10 percent thick- _z ness, like the NACA 0009 or the FX100 'I ergy storage; this means a flight speed of -[ 7 9.7/ 54 km/h. A new speed record could be- MP, are favorable as they work well even set with such an airplane. at low Reynolds numbers (below .5,EIR NGs _ X 300,000). Although elliptical tail wings _1 /N are aerodynamically better the G Wing shape and structure trapezoid R2. shape is advantageous as it can be built Ia P Though an elliptically shaped wing easier and lighter. I.7_/ seems aerodynamically the best, a trape- When arranging the elevator, great care 25 3 ; _ OP I JH Ilf a. FA zoid wing is preferred for design and aero- has to be taken that it avoids the down- i 7Vl ,LA wash turbulence of the wing at all flight 71 tl' dynamical reasons. A trapezoid wing can a. LA - ,ul , ,l ,W"`| ....,""..... YIY l .I be built to be simpler, lighter in weight altitudes. I Mrru (..I Ii and more accurate in shape, and the air- FIG. 4; Velocity polar of man-powered foil operates better at the wing tips be- Fuselage planes with optimum wing configuration cause of the larger chord length and high- HUMAN POWER, I Spring,~ I I 1987 vol. 6 no. 1 page 7

I GF/CONTICELL/GF CF-UO UPPER CHORD A DAY FLY a few centimeters off the ground for a few seconds on Dec. 6, 1985. As the power requirement for get- ting out of ground effect is more than 2 kilowatts, it does not seem to be likely that a general break-through will be achieved in the near future. The precise design of high-strength mi- D crolight construction developed for HPA MYLAR FOIL STYRODUR RIB CF-UD TEN51UN unuru IKAILIN6 UIE in the past years spurred work in the unu- COVERING sual aerodynamics at Reynolds numbers FIG: 5: Wing structure with Wortmann Airfoil FX 76 MP 160 between those for model aircraft and there for gliders. light-weight and reliable when carefully When , the pilot has only to The technology developed will also designed and built. In HPA-projects in imagine that he holds the wingtips with benefit the unmanned high-altitude air- Japan and Germany it was found that his hands and moving the controls will craft powered with solar orhybrid energy, the pedal-power output could be im- cause the plane to perform the desired used for example for communication re- proved by about 5 percent through the maneuvers. lays which can remain aloft in the strato- use of an elliptical chainwheel. Sideway tilting of the control bar op- sphere for weeks or months. In order to protect fuselage and erates theailerons, rotation around the wings from the accelerated and turbulent vertical axis acts upon the main rudder, E. Schoberl Ossiacher Str. 42 slipstream of the propeller, it was often and tilting the handgrips around the hori- D8500 Nurnberg 50 located behind the wings (MacCready's zontal axis acts on the elevator. As the West Germany BIONIC BAT) or behind the tail plane. aerodynamic control forces are rather (Rochelt's MUSCULAIR 1 and 2). small, it is practical to install self- Although a long propeller shaft is ne- centering spring units. cessary a pusher propeller is advanta- For long-distance flights it is desir- geous, as the tail plane works more ef- able to install an automatic control sys- fectively in the accelerated air flow be- tem to reduce the pilot's mental work- fore a propeller, and the power demand load. Such an autopilot can improve the of the HPA can be slightly reduced. system performance by maintaining flight near minimum power require- Controls ment. For the Daedalus project MIT de- Most HPA designers have arrived at signed an autopilot that weighs only a

elegant andergonomic control. Because few hundred grams, so that the pilot can Id precise control with steering handlebars concentrate more on power performance rE (fig. 7 shows the MIT-design) is very and navigation. ON important, the pilot must keep his/her -3 W body almost immobile above the hips Technological outlook a- to allow the controls to be handled sen- sitively, while with high power MIT may be making the flight legend output. of Daedalus a reality in the near future. This may be the culmination of human- powered flight which will remain a priv- ilege to a few enthusiasts and universi-

AILERONS ties only, because it requires both a highly sophisticated technology and RUDDER high athletic and flying effort of the pi-

C lot. - The development of human-powered FLYING SPEED [m/s] FIG: 6: Propellerdiameter requiredfor 85% VATOR helicopters is, because of the high power and 89% efficiency. requirements and the stability and con- trol problems, very difficult.

Nevertheless, Professor Akira Naito's team from Nihon University Tokyo did FIG. 7: Three-axis control stick. in fact succeed in getting their helicopter papVDaye 8 HUAHUMAN PnWER.OE.Srn. SItrin-. 1983797vl vol. 6 no.o I continued from front page A very simple estimate of the speed attainable can be Many, if not all, of the new high-speed craft will be made based on the propeller and drive-train efficiency, r, the hydrofoils. By literally flying through the water, hydrofoils power-to-weight ratio, P/W, and the lift-to-drag ratio, L/D: eliminate the water drag of a hull. The required wing area for a fast hydrofoil is quite small - on the order of .5 square V = rl P/W L/D feet or less. This paper will discuss the design of human-powered hydrofoils, concentrating on wing sizing based on power available, minimum flight power, takeoff Assuming an efficiency of 80% (.8), power of 0.5 HP (550 speed, and structural considerations. ft-lb/sec) weight of 200 lb. and L/D of 40, the predicted speed is 44 fts, or 26 knots. These are approximately the values for There are many hydrofoil configurations suitable for the Rochelt Musculair II human-powered airplane. The L/D human-powered applications. Here I will concentrate on the of 40 is typical of a modem sailplane (L/D is equivalent to the glide ratio). but is extremely Flying Fish configuration (Figure 1) - a single main wing good for a human- powered under the center of gravity and a small forward stabilizer airplane. Hydrofoils have sources of drag related to the wing which carries little or no load. The main wing is air-water interface that airplanes do not have, and hence have lower supported at its center by a single strut which houses the drive L/D values than airplanes. These differences will be explained mechanism to the propeller. Other more complex in greater detail below. configurations (ladder foils, biplanes, multiple support struts, Even if a hydrofoil L/D of only 20 could be achieved, etc.) may prove to be better for really fast designs. The 13 knots would be possible on 0.5 HP, and 26 knots on 1.0 analysis ideas presented here can be readily extended to these HP. It should be noted that for a given hydrofoil craft, the cases. L/D is a function of speed, and decreases with increasing Hydrofoil History speed above a certain speed. It is also more difficult to make a hydrofoil with L/D of 20 at 26 knots than at 13 knots. Brad Brewster [1] designed a human-powered While it is true that a wing alone optimized for 26 knots can hydrofoil as a bachelor's thesis project at MIT, but did not have the same L/D as a wing optimized for 13 knots, the drag pursue development because the V-foil configuration was of the rider and framework in the air is higher at 26 knots, predicted to have only moderate performance. I-le concluded resulting in a lower overall L/D. High-speed hydrofoils can that tandem fully-submerged foils were the best approach, certainly benefit from lightweight aerodynamic fairings but stability and control of this configuration would be a very similar to those used on land vehicles. complex problem. Jon Knapp experimented with hydrofoils on early versions of his Sea Saber, but never was able to get Induced and Wave Drag the hull completely out of the water, and concluded that the craft with hydrofoils had more drag than without hydrofoils. Like airplanes, hydrofoils have drag as a by-product of producing lift, called induced drag. Hydrofoil-induced drag James Grogono, a hydrofoil enthusiast in has the same functional form as airplane-induced drag, but England, developed the first human-powered hydrofoil craft there is an additional multiplicative factor, fi, to account for to successfully 'fly' on its foils [2] - a rowing shell equipped the proximity of the wing to the water surface. with a central fully-submerged foil, and a V-foil in front - but it too had more drag with foils than without. David D = q Cdi S Owers, also in England, has been developing Foiled Again, a i outfitted with hydrofoils and propeller drive, for the or last few years [3], with moderate success. = f W2 / q b 2)

The author and Allan Abbott have recently developed 2 the Flying Fish and Flying Fish II hydrofoils [4,5]. The Cdi = fiCL / A configurations of the Flving Fishes are quite different from previous designs. They incorporate tandem submerged foils, CL = W /(q S) active depth control, and a standard cycling position. The original Flying Fish, powered by Steve Hegg, now holds the q = 2 flying-start 2000-m record with a time of 6:38 (the 0.5 pV standing-start rowing record is 6:49). Here, p is the density of vwater, V is the velocity, q is the Hydrofoil Drag, Propulsion, and Performance dynamic pressure, W is the total weight, Cd i is the induced- drag coefficient, CL is the Hydrofoils vs. Airplanes wing-lift coefficient, S is the wing area, b is the wingspan, and A is the aspect ratio ( b2 /S). A hydrofoil is essentially the same as an airplane, but with the wings in the water. Water is more than 800 times Hoerner [6] gives more information on the factor fi. If more dense than air so we might expect larger drag. the wing is far below the water surface, fi is equal to one, However, since lift and drag are both proportional to density, the overall drag of a hydrofoil wing should be no higher than giving the same induced drag as an airplane. As the wing for an airplane wing. The hydrofoil wing has 800 times as approaches the surface, a sort of 'reverse ground effect' occurs, increasing induced drag because the wing has less much drag per unit of area, but it needs only 1/80 0 th the area of the airplane wing. fluid available to act upon. When the wing is at the surface, the fluid volume around the wing is halved, and fi reaches a limiting value of 2. HUMAN POWER, Spring, 1987 vol. 6 no. 1 _ page_ 9

Unlike airplanes, hydrofoils can produce waves which could be enough to make a significant difference in record cause an additional drag component, although experience trials. The following table illustrates this phenomenon for a with the Flying Fish suggests that wave drag is very small. 3-inch chord section at 12 kt. Except at very low speeds, near stalling, the visible waves from the Flying Fish are very small. Hoerner [6] also reports Temp k/P Re section drag that shallow water can reduce or eliminate wave drag. In (deg F) ft2 /sec relative to 5 \---- / fact, wave drag vanishes at a critical speed, V = (g H)- , -5) where g is the acceleration of gravity and H is the water ( x 10 70 deg depth. At a typical hydrofoil speed of 12 kt (20.2 ft/s) the depth required is 12 ft. At much lower depth, wave drag is 50 1.41 359,000 1.10 probably still negligible, and the induced drag may be 60 1.22 415,000 1.05 decreased due to ground effect as the wing nears the bottom. 70 1.06 477,000 1.00 80 .93 544,000 0.96 Profile Drac 90 .83 610,000 0.92

Profile drag is the drag of the wing and strut sections, A rule of thumb is that the profile drag coefficient decreases and is of the same form as that for airplanes: one percent for every 2 deg F increase in water temperature. For simplicity, Reynolds-number effects will not be included Dp = qCdo S in the examples below. If Reynolds-number effects were included, the optimum spans and aspect ratios would be where Cd o is the profile drag coefficient. S is the reference slightly smaller (i.e. larger chord), driven by the decrease in area (usually span x average chord). profile-drag coefficient with increasing chord.

Selection of the proper foil section is critically Interference Drag important to achieving optimum performance. Circular-arc foil sections as used on large powered hydrofoils are not Interference drag is caused by the mutual interference appropriate for human-powered applications. These are of wings and struts at an intersection. There is no easy way to designed to minimize the pressure variation on the upper calculate interference drag. Hoerner [6] presents some surface, and thus minimize the possibility of cavitation. They estimation techniques, based on various test data. For a basic do not have particularly low profile drag. Human-powered 'T' intersection of two foil sections, Hoerner gives the hydrofoils are slow enough that cavitation is not a concern, so following: the foil can be a standard airfoil as used on airplanes. For practical hydrofoil designs, it is usually desirable to have a Dint = qCdint t2 low takeoff speed, hence high CLmax, and low drag at high 2 speed, hence low Cd o at low values of CL. These Cdint = 17 (t/c) -0.05 requirements are the same as those for sailplanes, which must circle slowly in thermals, and fly at high speed between where t is the average thickness of the intersecting struts, and thermals. tic is the average thickness-to-chord ratio. This formula is only a rough guideline. Addition of good filleting, or The Reynolds numbers for hydrofoils (200,000 to staggering of the intersecting foils in a streamwise direction, 1 million) are too low to make use of full-scale sailplane can greatly reduce or eliminate interference drag. A airfoils. (Reynolds number:a measure of the ratio of inertial practical solution for a human-powered hydrofoil is to forces to viscous forces: Re - V c p / , where c is the wing- cantilever the wing forward from the vertical strut. chord length, and A is the viscosity). Luckily, these Reynolds Spray Drag numbers are a good match to those of radio-controlled model sailplanes. Especially applicable is the F3B competition The vertical struts which support the hydrofoils create a category, in which one of the events is a high-speed run. F3B spray of water at the point where they pierce the water competition has become increasingly fierce in recent years, surface. It might be expected that spray could be reduced if and considerable analytical and experimental effort has gone the strut section has low t/c and a sharp leading edge. into airfoil design. The HQ airfoil family, designed by However, experience with a variety of strut shapes for the Quabeck [7,8], appears to have the best overall performance. Flying Fish showed that the amount of spray is largely These are a family of airfoils of differing thickness and independent of the leading edge shape. Hoerner's [6] camber. discussion of spray drag indicates that it is only a function of the thickness of the strut: Airfoil profile drag coefficients generally decrease with increasing Reynolds number. For model airfoils between Dspray = .24 q t2 Reynolds numbers of 200,000 and 600,000, the drag - 3 5 coefficient varies approximately as R . Obviously, Thus the spray drag is roughly equal to the drag of a performance will be better if, all other things being equal, t/2-by-t/2 flat plate aligned perpendicular to the flow. On the Reynolds number is increased. A property of water is that its Flying Fish, the spray originates at the leading edge of the viscosity decreases with increasing temperature. Thus strut, and climbs about 8 inches up the side of the strut as a Reynolds number, and hence drag, can be minimized by sheet as it goes back. The added 'wetted area' of the strut operating in warm water. The drag reduction is modest, but results in additional skin-friction drag. HUMAN PnWER. Sring%19837 vol. Lpae 10 H.r .1 . 6 nn. 1

Air Drag The following table lists the values of the variables used in creating the graphs in the following sections. It is usually not necessary to consider air drag on a human-powered water craft, as the speeds are quite low. But symbol descrintion value units in going for all-out speed, say 20 knots, air drag becomes symbo - e-rin'riY'' density of water 1.938 quite significant, so aerodynamic streamlining should be P slugs/cu-ft added. Drawing on the analogy of hydrofoils and airplanes, a Pair density of air 0.00238 slugs/cu-ft hydrofoil without aerodynamic streamlining is akin to a CLmax max wing lift coef. 1.1 human-powered airplane without fuselage streamlining. Of W weight course, aerodynamic streamlining on watercraft has many 190 lb. t/c thickness ratio practical problems, foremost of which is safety. Watercraft, .13 especially hydrofoils, can and do capsize. A rider inside a k taper ratio .4 streamlined enclosure may find it very difficult to escape propeller efficiency .85 while submerged. Partial streamlining such as handlebar Tlprop fairings, streamlined tubing, and special clothing is much fi ind. drag factor 1.48 more practical. Cd o wing profile drag 0.008 strut profile drag 0.0085 Propulsion Cdstrut 2 Sstrut strut area 0.46 ft Virtually all of the new generation of high-speed water Cdfw front-wing drag 0.009 vehicles will be driven by propellers. It has been known for a 2 long time that propellers are more efficient than (albeit Sfw front-wing area 0.43 ft less practical in weedy lakes or shallow water). Efficient Cdspray spray-drag coeff. .24 propellers for human-powered craft are quite different from tstrut strut thickness .0917 ft propellers used on engine-powered . Slender blades and steep pitch are the rule for human-powered applications. One Cdair air-drag coeff. .7 of the 2 first applications of such propellers was on Calvin Sair frontal area in air 7 ft Gongwer's human-powered submarines in the mid-1950's. More recently, propellers optimized using the methods of Design Power Larabee [9] are used on craft such as the Knapp Sea Saber, the Owers Foiled Again, and the Brooks/Abbott Flying Fish The starting point for design of a human-powered series. hydrofoil is the design power level. It is determined from the duration of race or event for which the hydrofoil should be It is possible to get propulsion directly from a hydrofoil optimized. Of course the duration of a distance event depends wing if it is flapped up and down. This approach has high on the expected speed - some guesswork or iteration is efficiency theoretically, but is quite difficult to implement. required in these cases. There are published curves ( see e.g. The first known flapping hydrofoil craft is seen briefly in the Whitt and Wilson [12] ) which show the attainable power movie "Gizmo!". It consisted of two hydrofoil wings, one output of various types of humans (from average people to for each foot, with handholds to aid in pitch control. The world-class athletes) for different durations of exercise. For 'rider' stands sideways to the direction of flight and flaps the the design examples in the next section, the following three wings with arm and leg motions. Nothing is known about the combinations of power and duration will be used: designer or origin of this craft. Bill Watson recently made a reproduction [10], but it proved to be impossible to control. Duration PoePower A .nnnlictinn _c L ti A Parker MacCready, an engineering-science graduate student Duration~~~~~~~~~~~~ at Caltech. is now researching flapping hydrofoils [111]. He 40 seconds 1.0 HP 200-m event recently completed a testbed flapping-wing vehicle with a 6 minutes 0.5 HP 2000-m event configuration similar to that of the Flving Fish. Ihr 0.25 HP recreation

Hydrofoil Design Considerations Maximum Speed

In this section. the functional relationships of the The maximum speed occurs when the design power, various design parameters will be given. Recall that the reduced by the propeller efficiency, is equal to the power designs in this paper will be based on the Flying Fish required. The power required is the product of the velocity configuration. The goal of the design process will be to select and the sum of all of the drag components. For simplicity, all the optimum wingspan and aspect ratio, based on several of the drag terms which are not related to the wing are design criteria. The main criteria are : speed at design summed to give an equivalent 'drag area'. Sref . (The air drag power. minimum power, takeoff speed, allowable bending term is mulitiplied by the air-to-water density ratio before stress, and wing-tip deflection. In order to easily see how summing). The value of Sref based on the data in the above these factors interrelate, graphs will be given where constant 2 v alues of the factors (contours) are plotted on axes of aspect table is .0158 ft . ratio and span. By overlaying these plots, insight can be ailned into various tradeoffs (e.g. how much can speed be Pdesllprop = V { qCdoS qSref fW 2 / (qb-, increased by going from a fiberglass wing to a carbon-fiber } wing?). 3 2 2 2 = .5pV (Cdo b /A +Sref) + fiW / (.SrpVb ) HUMAN POWER, Spring, 1987 vol. 6 no. 1 page 11

Minimum Power It is seen that the takeoff speed varies as the square-root of the wing loading. With a wing loading of 130 lb/ft2 , the Flying A hydrofoil optimized for maximum speed at a given Fish II is able to take off easily. Assuming a CLmax of 1.1, power may have quite high minimum power required just to the corresponding speed is 7 knots. Contours of takeoff fly. This may be acceptable for an all-out racing machine, speeds between 4 and 12 knots are shown in Figure 3. but for versatility or recreational use, it may be desirable to be able to fly at a lower power. Specification of minimum 35. power as well as design power places a constraint on the allowable designs, and usually will result in a penalty in speed 30. at the design power.

c- 25 Minimum power is found by setting the derivative (with ¥ respect to velocity) of the power equation equal to zero. This L 20. leads to: 0) 4 2 2 W 2 2 2 15 3 {.5pV (Cdo b /A +Sref)} = fi / (.5rrpV b ) 10. which can be solved for V, and then V substituted into the original power equation to get the minimum power level. 5. Figure 2 shows contours of minimum power from .15 HP to 2 3. 4. 5. 6. 7. 8. 9. .35 HP. (The contours were calculated by iteratively Span, ft. calculating values of A that resulted in minimum powers that matched the contour levels). Figure 3. Takeoff speed contours, assuming takeoff at CL= 1.1 35.

30. Bending Stress

0 25. The bending stress at the root of a thin high-aspect ratio C L hydrofoil wing can be very large. (It will be assumed that the U 20. hydrofoil wing is constructed as a solid mass of material; no 0.3 lightweight core, etc). From simple beam theory, the <[ 15. maximum bending stress, , is

10. = M/Sm

5. 2. 3. 4. 5. 6. 7. 8. 9. Where M is the bending moment, and Sm is the section Span, ft. modulus. Assuming an elliptical distribution of lift along the span, the resultant lift acts 42% of the semi-span out from the Figure 2. Contours of minimum power required to stay root. This leads to a root moment of: foilborne as a function of wingspan and aspect ratio. M = .11Wb

The section modulus is a function of the thickness, chord, and Takeoff speed distribution of thickness. For the HQ family of airfoils the section modulus is: Like the minimum power case above, hydrofoils optimized only for speed at high power may prove to have an 2 3 uncomfortably high takeoff speed. Takeoff occurs when the Sm = 0.0746 (t/c) c wing generates lift equal to the total weight: The factor .0746 was calculated numerically for the HQ W = CLmax .V5 2 S family of airfoils. It doesn't change more than about 10% for other airfoils. After including the effects of wing taper ( is Solving for V: the ratio of tip chord to root chord):

2 3 3 2 2 } V = 2 W / (p S CLmax) }1, o .11 W A (1+ ,) / 8*.0746 b (t/c) Note that taper ratio and thickness ratio have a strong 2 = { 2AW/(pb CLmax) }l/2 influence on the root stress. It is especially beneficial to taper .page 12 HUMAN POWER, Sring,1I I 1987 vol. 6 no. 1

the wing. Although an untapered, constant-chord wing may where E is the modulus of elasticity of the wing material. be easier to make, a tapered wing has much lower stress, and The factor .037 is constant in the wing moment of inertia: I = is better hydrodynamically. For an untwisted, .037 (t/c) 3 c4 and is accurate for the HQ family of sections and straight-tapered wing, a taper ratio of .4 is about optimum is reasonably close for other airfoils. The function f, of taper for achieving a nearly-elliptical lift distribution. (Twist is and load distribution, is best calculated by numerical not desirable if the wing must operate over a broad range of integration of the bending equations. Figure 5 shows lift coefficients). A thickness ratio of 13 percent is a contours of tip deflections between 4 and 20 percent of reasonable compromise between low drag and structural semi-span. Figure 6 shows the effect of the material stiffness. considerations. Using these values, contours of root bending Lines are shown for wood, fiberglass, aluminum, carbon, stress between 5,000 psi and 25,000 psi are shown in steel, and HM carbon (special high-modulus carbon), for Figure 4. equal deflections of 8 percent of semi-span. 35 35. 30. 30

25. 25 20 -5Z 20. -1 -: 15 15.

10 10 5. 2. 3 4. 5 6. 7. 8. 9 5 Span, fi 2 3. 4. 5. 6. 7. 8. 9. Span, ft.

Figure 4. Root bending stress contours, assuming taper Figure 5. Contours of tip deflection normalized by ratio of 0.4 and foil thickness/chord of 13%. semi-span (b/2), assuming taper and thickness as in Figure 4, with material properties of aluminum or wet-layup carbon fiber. Tip Deflection

High-aspect-ratio hydrofoil wings can have considerable 35 deflection at the tip under load. Both translational and torsional deflections are important. If the wing is too 30 flexible, the wingtips may rise up out of the water, or 'flutter' may occur at high speed. Flutter is an aeroelastic (or 25 hydroelastic) oscillation. In airplanes, flutter is often a) disastrous, tearing the airplane apart. Flutter probably 20 but 7"4 wouldn't structurally harm a human-powered hydrofoil, 1±1 it would ruin any chance of going fast, due to a large amount 15 of extra drag. Flutter calculations were not made for either Flying Fish. The wings for both 'Fishes' were designed with 10. a somewhat-arbitrary tip deflection limit of 8 percent of semi-span, i.e. about 3 inches, and flutter has never been a 5. problem. 2 3. 4. 5. 6. 7. 8. 9 Span, ft. Torsional deflections alter the spanwise lift distribution, causing increased induced drag. Torsional deflection Figure 6. Effect of different wing materials for depends on the torsional stiffness of the wing. sweep angle, normalized tip deflection of 0.08. the airtoil piitching moment, speed, and location of the shear center of the wing structure. Torsional deflections are beyond the scope of this paper. It suffices to say that torsional Design Examples deflections shoud be kept very small. The graphs of the previous section can be used to evaluate the various tradeoffs involved in selecting wing span The normalized tip deflection, A = 6/(b/2) can be and aspect ratio. It is not generally necessary to consider calculated using elementary beam theory. The result is: stress limits if deflection limits are also applied. The stresses at maximum deflections are almost always quite modest and A = 6/(b/2) = f(.,loading) W A4 / (64 .037 E (t/c)3 b2 ) are well below the ultimate or yield strengths for the material. TJITAAANT DnTAIFP Cnrncr 197 xrnl nn 1 page 13 I I CLVIZI 1 I / V J -, _, F11 lb/ I- - L· -·r - 35. be Three basic designs will be considered. All will 30. designed initially to have a minimum power requirement of 0.2 HP to allow extended-duration cruising, and maximum .o 25. normalized wing deflection of 0.08. The wing construction .0 material is assumed to be either aluminum or carbon L 20. fiber/epoxy. The first design is for a recreational craft, with -W Cn a low design power of .25 HP and maximum takeoff speed of 0 15. only 4 knots. Second is a competition craft designed for 2000-m races, with design power of 0.5 HP and maximum 10. takeoff speed of 8 knots. Third is a racer for short 200-m sprints, with design power of 1.0 HP and a maximum takeoff 5. speed of 10 knots. 2. 3. 4. 5. 6. 7. 8. 9. Span, ft. Figure 7 shows contours of maximum speed attainable for the recreational design. Constraint lines for takeoff speed, minimum power, and deflection are also shown. Figure 8. Maximum-speed contours for the 2000-m racer Taken together, the constraint lines form the boundaries of a (0.50 HP), with constraint lines. region, inside of which all constraints are satisfied. The best span and aspect ratio combination inside the region is that Figure 10 shows the optimum planforms to scale for which is on the highest maximum-speed contour contained each of the three wing designs with the original constraints within the region. If necessary, rough interpolation can be and without minimum-power and takeoff-speed constraints. used to estimate the shapes of intermediate speed contours. It The following table gives a summary of the constrained and is seen that the best design point for this case has a maximum unconstrained speeds for the three designs. speed of about 7.6 knots, with a span of 9 ft and aspect ratio of 20 (the actual optimum is just off the edge of the graph). If Design max speed max speed w/o min pwr the takeoff speed and minimum power constraints are (knots) or takeoff contstraints removed, the maximum speed increases to 8.4 knots, the span decreases to 6.5 ft., and the aspect ratio increases slightly to recreation 7.6 8.4 22. This design still has a relatively modest takeoff speed of about 6 knots (this can be seen by overlaying Figure 3 on 2000-m 11.7 11.8 Figure 7 or by tracing additional takeoff speed contours onto Figure 7). 200-m 15.3 16.0 35. 35. 30. 30. o 25. 25. 20. .) () 20. a) 15. C,) 15. 10. 10. 5. 2. 3. 4. 5. 6. 7. 8. 9. 5. Span, ft 2. 3. 4. 5. 6. 7. 8. 9 Span, ft. Figure 7. Maximum-speed contours for the recreational (0.25 HP) design, with constraint lines. Figure 9. Maximum-speed contours for the 200-m racer (1.00 HP), with constraint lines. Figure 8 shows the results for the long-distance racing a maximum speed of 11.7 craft. The optimum design has The speed of the best 200-m sprint craft is still below the In this case, minimum power and deflection are the knots. 20 knots mentioned in the title of this paper. To increase limiting factors. If the minimum-power and takeoff-speed speed further, the wing material could be changed to constraints were removed, the speed would increase only high-modulus carbon or steel to allow a higher aspect ratio, slightly to 11.8 knots (5 min:32 sec for 2000m). but this alone isn't enough. The drag of the craft must be reduced as well. With the addition of an enclosed fairing, air Results for the sprint craft are shown in Figure 9. The drag could be cut in half. The main vertical strut could also optimum design has a maximum speed of 15.3 knots. be made smaller by about one-half to reduce spray and skin- Without the minimum-power and takeoff-speed constraints, friction drag. The area of the front wing could be made the speed increases to 16.0 knots. smaller, also by half. Figure 11 shows the results of these rape 14 TJ1TA/fAAT pf)lAIRP ,,;,,r 10Q7,,-.1 -l -,-. !F-__---) 1I%-VYL1, )W IL-,I; / V 1 O *r'~r Planform with constraints on mn power, takecf seed. and eflection The 20-knot hydrofoil is certainly a possibility with With constraint on deflection only today's technology. With the encouragement of the IHPVA, and possibly a prize patterned after the Dupont Prize for land ~i-- Y s1. ' I vehicles, the 20-knot barrier will surely fall. 0----- 35. Recreational design: 0.25 HP

34 30. 7 - -

0 25. I= II~NMOE~ a,._ . 2000-m-event design: 0.5 HP 11 7 t ma, 20. r 11 8 in 15.

10. 200-m-event design. 1.0 HP \ , 15 3 kt max 16 0 5. 2. 3. 4. 5. 6. 7. 8. 9. Figure 10. Optimum wing planforms for three optimal Span, ft. designs. Figure 11. Maximum-speed contours for the ultimate 200-m changes along with a deflection-limit line for high-modulus racer (with reduced air, strut, and front-wing carbon or steel. 20.2 knots can be reached by this ultimate drag), with deflection constraint for craft with a wingspan of only 2.5 ft and aspect ratio of 18. high-mnodulus carbon or steel. The takeoff speed of 13 knots may prove to be an inconvienience, however! A larger wing with just 20.0 knots top-speed capability would have a slightly more reasonable 1.2 takeoff speed of 11 knots. 1.0 Figure 12 shows a comparison of the power vs velocity requirements I for the three original hydrofoil designs and the "- .8 final 'ultimate' 20-knot craft. Also shown is the curve for a single rowing shell, based on the data of Rogen [13], and .6 assuming that the efficiency of rowing is 70 percent. Note a) that at low speeds, displacement hulls are as good as, or better .4

than hydrofoils. It doesn't pay to make a hydrofoil unless you CL want to go fast and pedal hard! -_ Recreationalhydrofoil .2 ...... 200 : 2000m hydrofoils Ultimt hydrofoil Rowingshell (Rogen test data) .0 Conclusions 0. 5. 10. 15. 20. 25. Velocity, knots The examples in the previous section have just begun to scratch the surface of the kinds of tradeoffs that are possible. Figure 12. Power vs. speed for the design examples, and for Many more questions and tradeoffs can be explored by a single rowing shell. Note that 2000-m and overlaying the graphs or tracing lines from one graph to 200-m designs are the same due to the minimum- another. Questions such as: How much is top speed reduced power requirement. by having a low takeoff speed? How much faster at 1 HP is a craft optimized for 1 HP than a craft optimized for .25 HP? Alec What is the tradeoff in top speed by setting a low minimum N. Brooks required power? Is it worth the extra money to make the 1536 N. Altadena Dr. w ing out of carbon fiber instead of wood or fiberglass? Pasadena, California 91107 USA

References 10. Watson, B., "Current Trends in Water Power," 1. Brewster, M. B., "The Design and Development of a 5. Brooks, A. N., "Waterbike!," Bicycle Guide, Vol. 2, Proceedings of the First Human-Powered-Vehicle Man-Powered Hydrofoil," Thesis for Bachelor of No. 3, May, 1985. Scientific Symposium, Science, Mechanical Engineering A. Abbott, ed. IHPVA, Seal Dept, M.I.T., May Beach, Ca. 1982. 1979. 6. Hoerner, S. F., Fluid Dynamic Drag, Hoerner Fluid Dynamics, Brick Town, N.J. 1965. 11. MacCready, P., "Features of Flapping-Wing 2. Grogono, J. L., "Human-Powered Flight on Propulsion," Proceedings of the Hydrofoils: A Successful -Craft Conversion," 7. Quabeck, H., HO profile fur F3B. Thermik. Han. Third International Human-Powered-Vehicle Scientific Symposium, A. Proceedines of conference Elektro- und Gross-Seeler, Verlag fur Technik und on Human Powered Marine Abbott, ed. [HPVA, Seal Beach, Ca. 1986. Vehicles, Royal Institution of Naval Architects, Handwerk GmbH, Baden-Baden, 1983. London, 1984. 12. Whitt, F. R. and 8. Althaus, D., Profilpolaren fiir den Modellflue Wilson, D. G., Bicycling Science, The 3. Owers. D. J., "Development of a Human-Powered volumes I and 11, Neckar-Verlag, Villingen- MIT Press, 1979. Racing Hydrofoil," Human Power, Vol. 3, No. 3, Schwenningen, West Germany, 1985. 13. Rogen, B., "Determination of Drag Force Spring 1985. 9. Larabee, E. E., "Propellers for Human- Powered Relationships and a Drag Coefficient for a Rowing 4. Brooks. A. N., "The Flying Fish Hydrofoil," Human Vehicles," Human Power, Vol. 3, No. 2, Winter 1984. Single Scull," Proceedings of the Second Intemational Power, Vol. 3, No. 2, Winter 1984. Iluman-Powered-Vehicle Scientific Symposium, A. Abbott, ed. II-PVA, Seal Beach, Ca. 1982. HUMAN POWER, Sprim-;, 1987 vol. 6 no. 1 vage 15 DYNAMICAL STABILITY OF THE BICYCLE

By Y. LeH6naff

Editor's note: The following paper was sent in by a noted Frenchnuclear physicist. Having had a long series of weighty papers on nuclearphysics accepted without demur by various journals,he ventured into the clouded waters of bicycle stability - and had his paper rejected. I created a small storm of controversy when I used some of David Jones' article in PHYSICS TODAYfor BICYCLING SCIENCE (I confess I also introduced an error)so thatI knew of the lack of agreement in this surprisingly X complex field. I wrote to the author to I , 'I I . I state that HUMAN POWERwould be FIGURE 1: Side view and bicycle demensions. Lengths: MM'=TT'=MP + PM'= 1 + honored to publish his paper, but would L = a =1.0 m; 1 = 0.2 r; wheel radiir =0.33 m. T=TO is measured positively invite comment and a rebuttal. Here are forwards. The fork angle is x= 20 . all three pieces. - Dave Wilson When the system "rider + bicycle" runs at the frame plane. We write MP -- 1, PM' ABSTRACT speed v on a circular path, the rider leans L and, of course, 1 + L = a. The first dynamical stability curves with the frame tilted by an angle 0 from for bicycle riding are obtained as the vertical into the curve to offset the The angles , - (POX) and - solutions of two coupled non-linear centrifugal force. The steering angle S is (POX') are defined respectively in the equations. the dihedral angle between the plane of front-wheel and frame planes; only when the frame and the plane of the front those two planes coincide do we have V Introduction wheel, whereas the angle of turn Cx is the + = - ' = 1t/2. When the handlebar is intercept of this dihedral angle by the turned, it can be seen experimentally that The theory of bicycle stability has ground plane. If a is the distance between the front wheel slides slightly been a long-standing problem since 1898 the wheels' contacts with the ground, the downwards; therefore, , increases and a [1] and culminated with the works of E. radius of the curve is roughly R = a/tanCx. decreases. 2 Carvallo [2], F. Klein and A. With g = 9.8 m/s , the system is in Sommerfeld [3], all using small-angle dynamical equilibrium when: Jones computed the height h of the approximations. The most fork point - which is related to the height comprehensive simulation of bicycle tan0 + v2 tanC< (ag). [1] of the center of gravity and hence to its riding was done by R. Douglas Roland potential energy - h = HP, in the frame [4] and involved the computer resolution According to tradition, "stability" is plane for obvious dynamical reasons. of eight simultaneous differential meant here by "equilibrium". Other Writing OP - Y, we have: equations with more than 70 parameters authors [6,7] have analysed steering to describe the "bicycle + rider" system. "stability" to front-fork vibrations that h = s n' (2) Using a geometrical approach, D. Jones are not considered here. In the frame plane, the vector [5] computed the height of the "fork 1 point" for an infinitely long bicycle, but Geometry of the bicycle relation OP + PM + Hi]T + ¥V0 = O failed to account for the stability. projected on M'T' yields: For the upright bicycle in Figure 1, The present article solves the the plane of the front wheel coincides >Ys; n + L sin ( + - T/2) = r, (3) problem for a normal-length bicycle with with the plane of the frame and cuts the where r is the radius of the wheel and two coupled non-linear equations, derives ground plane along XX'. M and M' are the first dynamical-stability curves for the centers of the hubs, T and T' the (4r + - /2) is the downward tilt of the bicycle riding and analyses the trail effect ground-contact points of the wheels. The frame in its plane. on steering. front fork axis makes an angle &with the Similarly, in the front wheel plane, vertical, intercepts the ground at O and the corresponding relation P + PM + T0 Gravitation and Centrifugal cuts the line MM' at P, defined as the = O projected on MT gives: Forces fork point. P is fixed on the fork axis and projects at H on the ground trace OT' of x sin(T - X) + 1 sin(K - P - T1/2) = r. At usual velocities, equilibrium on a the frame plane: P is also at a fixed (4) bicycle relies essentially on the balance position with respect to the center of (. and X vary with the angle of of gravitational and centrifugal forces. gravity of the system supposed [4] within turn.) vaae 16 HUMAN POWER, Spring, 1987 vol. 6 no. 1

skillfully ridden, so that a realistic I potential curve should display a z I maximum characteristic of stable motion equilibrium, like the one obtained here for zero lean. Incidentally, the reason the other curves are not symmetric about 0C= 0 is simply due to the fact that with a constant positive lean 0 the steering leads either on the correct side (cx>0) or on the wrong side (x<0) of the lean. / O The observed minimums can be seen experimentally. At rest, when the frame is tilted, the front-fork assembly FIG. 2 Schematic diagram of bicycle on a left turn OX is the ground trace of the rotates into the lean to the front wheel plane and OY that of the frame plane, which is tiltedfrom the vertical corresponding minimum of the bike's plane (OY, OZ) by an angle 0. The front-fork axis is along OZ'. gravitational potential energy. But, as that Jones's solution with the rear wheel we shall see later, these minimums represent highly unstable riding set to infinity was fairly good. The Eliminating X from (3) and (4) conditions. provides the first relation between the remaining discrepancy is now accounted an error in Jones's calculation unknown angles V and )(: for by The conclusion is obvious: r + L cos4 cosp r + I cosX cosy [private communication]. - a sinq = 0 geometry alone does not provide the right potential curves. But can we s i not sinx However that may be, maximum ignore the main dynamical stability occurs at absurdly large angles, A second relation can be obtained contribution? considering a bike running on a curved as already noticed by Jones. We know the moving bicycle to be quite stable when path (Figure 2). We take 0 as the origin Dynamical Equilibrium of the coordinates; OZ' as along the fork axis; i,j,k,k' as unit vectors on axes OX, The gyroscopic forces on the OY, OZ and OZ' respectively; and finally wheels - on the order of a few newtons n as the unit vector normal to the plane for modem wheels - were studied by of the frame. hI Klein and Sommerfeld, who found them we have: sine = n-, = nxn sin*, negligible and unable to account for the iTx = k sina; giving: " tI an equilibrium. However, by inducing the correct centrifugal force, these forces do giving: contribute to the stability of the cosX + cosa cos - sine sina sin* = 0 (6) /o riderless bike and enable a rider to ride "hands off' - when the rider's weight * from the Eliminating ,- and H2 Z., lies mostly on the back wheel and the coupled equations (5) and (6) provides the reduced ground friction of the front relation we need between known wheel allows it to swivel freely. quantities (r,l,L,4) and given ones (Cx,0). c l _-~ -~ - leads to an intractable 8th-degree Algebra 0· However, the centrifugal force equation, but it can readily be solved by -3 I2 -I 1 . I0 itself can be taken into account by -3o -Zo -to 0 o 2o 30 5 iteration using a small computer. computing the height of the fork point subject to condition (1). The results of Figure 3 shows, for various lean the calculations are shown on Figure 4 angles 0, the relative height h/r of the E for different values of the speed v (with fork point in the frame plane plotted the same ordinates as above ) versus the curves versus the dihedral angle S defined FIG: 3: Geometrical-stability turn angle x. As expected, the curves previously. The curves also represent - Geometrical-stabilitycurvesfor the about Cx = 0 are (i) symmetric and (ii) ignoring constant factors - the potential bicycle of Figure1 are shown at different not too peaked; i.e., the instability is energy of the system. comparison is lean angles 0 in degrees. The height hlr not too dramatic, and fall can easily made with the results of Jones (broken of the fork point, in wheel-radius units, is checked by calling on the centrifugal line in Figure 3), using his conventions plotted versus the steering angle S in ° force through proper steering. and bike (,P= 20 , 1=0.2 r). The results degrees. For clarity, all curves have been Conversely, even though constant coincide exactly for 0 = 0 but do depart as arbitrarilyshifted vertically. Jones's ° attention is required by the rider to the lean increases. Running the program resultsfor 30 lean are shown in broken maintain the center of gravity close to again with a rear wheel 500 meters away line. Circles represent the values = 0, at 9 and 18 km/h (2.5 and 5 m/ obtained with the rearwheel moved 500 Cx does not change the results significantly s) the centrifugal force in a curve with (circle symbols in Figure 3), and shows meters to the rear. HUMAN POWER, Sprin, 1987 vol. 6 no. 1 I I I II ~Ac'olkn e d 17

Ak wackn ledge "useful correspondence with Prof. D. Jone's:andenlighte'ning discus'sions ,with Prof. A. 'Berroir and P. Vernin. "References

[1ngineerc w (London), vol.'30, December 21898.

[i21i~~;i;iT~e~ Ca valo E.,:Th:orie d mtrouvement du monocycle et de la -3o _20 _lo 0 fo o 3o ,< bicycletteUJurnal de1'Ecole: L~ ~~~3 2od o2 oo Polyt echnique, 2e ste, Cahiers5 etv FIG. 4: Dynamical-stabilitycurves Dynamical-stabilitycurvesfor the bicycle of Figure 1 are shown at different speeds, given in km/h and mis. The same ratio h/r is plotted versus the turn angle c in degrees. Again, the curves have been shifted verticallyfor [31 ,,Kl~in, F. andil-Sommerfeld, A., clarity. Constant lean angles 0 are shown in 'dashed lines, as well as the lean in the Teorie des: Kreisels Vo IV, ch. IX- minimums; these are of course symmetric about Oc = 0. 8Teub ner :,Leip si , ' 9 0 a o = 5 can be offset easily by a lean of further turning, as shown by the (negative) ° t4] D':oug1las Rola'nd R., C mputer, 3 and 120 respectively. This time, of increase of the second derivative. Simulatinof :fBicycle Dynamics, course, the lean angle increases either The consequences for steering-fork ::Calspan Corp, Buffalo, N.Y. with the speed or the turn angle, as it design are obvious. With a small trail (T should. = 20 to 30 mm), the bicycle will not lose t51Jne, D., The:'stab b'lility o)f'Thf the too much of the high responsiveness bictycl, Physics: Today p. 34, April' The stability curves become more particular to the zero trail, will still 1970. peaked as the speed increases but, due to benefit from the stabilizing caster effect, 2 the v dependence of the centrifugal and will have the dangerous minimums [,61 :Lowell, . and McKell, FL., The' force, a given lean angle can be offset by distant enough from cx = 0 for all practical stbi lity of bicycl es, Am. J. Phy. 50, a smaller steering angle, as shown by purposes. Even if empirical efforts have p. 1J106,Deember 1982. the dotted curve in Figure 4. This gives already led to satisfying results, the the feeling, which Jones sought to present theory may help to improve t7] ::Roc,1rd Y., h6oriegn6ale des:: explain, of greater stability at high design. As can be seen from Figure 5, ibrations 128 M on, Paris.1971 speed. Other stabilizing effects, such as when the trail length is kept constant, the gyroscopic forces, trail friction and the fork angle has practically no effect on V.e L n ff- :: :.:::: .. corresponding dampening of lateral stability. Consequently, builders may 49: rue de :·Chatenay-G3 oscillations do become significant [6, choose the fork angle subject only to Antony "92610-FRANCE 7]. mechanical constraints, and then choose the trail length according to the desired The trail T TO, as shown on steering flexibility. Figure 1, is: T = (r cos) + 1 cos )/sinX; it V. /,qk_ / 5· _j decreases on either side of c0= 0 and goes through zero at the minimums. The derivative, dh/d = T(cos sino< + sin0 sinm cos0)/ sin2Z; cancels either for T = 0 or for Cx = 0. This explains the stability around < = 0. However, the trail T changes sign at either minimum where the front wheel may then switch spontaneously backwards to the negative trail or "unridable" bike of Jones.

Figure 5 shows how the stability curves are modified around the equilibrium 0 = 0 by the trail length T: the longer the trail, the harder the rider FIG. 5: Variation of stability curves. Stability-curve variation is shown near the will have to hold the handlebars against equilibrium at O = , as afunction offork angle and/or trail length T. page 18 HUMAN POWER, Spring,! 1987 vol. 6 no. 1 ·Ill! ~ ~ ~ ~ ~ ~ POINTICOUNTERPOINTIPOINTICOUNTERPOINTIPOINTICOUNTERPOINTIPOINTICO UNTERPOINT S, plus gyroscopic terms. (Such modified Discussion of Le Henaff's Paper curves will be generally similar to those in figure 4. But in contrast to figure 4, the By Jim Papadopoulos and Surprisingly, for a range of speeds, many Andy Ruina "peakedness" of the curves - the 2nd bicycles are "stable" in this sense. derivative evaluated at S=0 - should Mathematical analyses and experiments probably not depend on rider speed v.) Even a bicycle in a steady turn, the both indicate, opposing some intuitions, subject of Le H6naffs interesting paper, Plotted in this fashion, for example, a that this "stability" is only possible for a bicycle with vertical head angle and non- is difficult to analyze. This is bicycle if its Le H6naff curves (modified immediately revealed by a glance at the zero trail would always make a "valley". as illustrated) are "peaked" (concave We appreciate Le Hdnaffs work on extremely complicated equations in ref. down) in the center. For such a bicycle in (2). Deep understanding can only come bicycle equilibrium and stability, and hope a steady turn, the handlebars need to be he continues his investigations. from the use of idealizations like those held against further turning! (However, if implied by Le H6naff: assuming a there is a rider, this torque is greatly References concentrated mass at the CG point; and altered when he/she bends sideways.) neglecting gyroscopic effects, The intuitive assumption that 1. Gray, Andrew (1918) A Treatise on deformation of the bicycle, and stability is synonymous with a potential- dissipative friction. Although Le Gyrostaticsand Rotational Motion; energy minimum does not apply to a republished by Dover in 1959. (See pp. H6naffs assumptions cannot be used to moving bicyle, which has finite kinetic 19-20 and 29-33.) predict all bicycle phenomena, they energy, gyroscopic forces, and non- 2. Man, G.K. and Kane, T.R., Steady seem appropriate for estimating and holonomic constraints [see refs (1), (3)]. Turning of Two-Wheeled Vehicles; in understanding the potential energy of One must be wary of stability arguments Dynamics of Wheeled handlebar torque in steady turns. that do not take into account the However, such quantities calculated Recreational Vehicles, SP-443, appropriate dynamical (time varying) Proceedings of 1979 SAE Congress in from steady equilibrium motions may equations of unsteady motion. Detroit, pp. 55-75. have only subtle connections to what Incidentally, to calculate handlebar 3. Neimark, J.I. and Fufaev, N.A. (1967) would be called "stability" - either by a torque during steady turns, probably Dynamics of Non-holonomic Systems; bicycle rider or by a student of the slightly different quantities should be translated from Russian and published in bicycle as a dynamical system (where plotted in Le H6naffs figures 4 and 5. 1972 by the American Mathematical the change of lean and steer-angle with The vertical axis should plot (the CG's Society, Providence R.I., (See pp. 330- time are governed by differential distancefrom the line between the front 374.) equations.) and rear-wheel contactpoints)l (the cosine Consider as a special case the of the angle of lean of the plane Cornell Bicycle Research Project dynamical stability of a bicycle with no containing the CG and the two contact Theoretical & Applied Mechanics rider. Call a bicycle "stable" if it points). The horizontal axis should plot Thurston Hall recovers to steady straight-ahead motion the steer angle S. The handlebar torque Cornell University after any small disturbance from that for a given steer angle S will be Ithaca, NY 14853 motion, and "unstable" otherwise. proportional to the slope of this curve at (607) 255-5035

The Author Responds...

· Whipple is certainly a good reference perienced them once, going down a slope * The reason I worked again the height if I judge by hearsay. Unfortunately, I with a loaded front fork; they are very im- of the fork point is two-fold, besides could not get a copy of Whipple's old pressive. the fact that it is a good way to get at paper. Can you send me a copy? Same - I agree, I mean equilibrium. And, ac- the c. of g. First, because Jones himself with Neimark & Fufaev's relevant pag- cording to Lagrange's theorem (in The was not sure of his calculations (cf. his es. Now, as for why my analysis, I Stability of Motion by N. Chetayev, subtitle!), inasmuch as it gave him thought it was plain from the subtitle Pergamon, 1961, p. 32), "if, in the equi- 'absurd' and 'unphysical' results - his of D. Jones' paper. Besides, Jim provid- librium position the potential function own words. His doubts stemmed from ed some of the answers between the has an isolated maximum, then this equi- the fact that he solved the geometry for brackets. librium position is stable." So, of an infinitely long bike (a fact I found * Front-wheel/fork vibrations have course, are the two minimums. However, only after my own calculation, con- nothing to do with equilibrium; they as I show later, the trail (of a non-zero- firmed later by him) and got wrong po- are related to what engineers call shim- trail bike) will not change sign at the tential curves. Second, I wanted to see my or wobble of wheels. These vibra- maximum, contrarily to what happens to whether I could get the correct curves tions obey two coupled differential the minimums with, eventually, dramatic with a normal-length bike. equations (see my ref. 7) in which the consequences on 'stability.' Now, the reason this height has to be trail has a dampening effect only. I ex- computed in the frame plane (in the plane of the bike, said Jones) is for a HUMAN POWER, Srin, 1987 vol. 6 no. 1 page 19 POINTICOUNTERPOINTIPOINTICOUNTERPOINTIPOINTICOUNTERPOINTIPOINTI CO UNTERPOINT

dynamical reason. When a pendulum is may turn the wheel around). Positive count here), that is excluding the degrees swept around on a circle, the dynamical- trail acts as a spring to straighten the of freedom of the rider himself, for which ly stable position is when its c. of g. is front wheel of a forward-going bike; no classical equilibrium theories from La- lowest along the supporting line. i.e. in wonder a negative spring will compen- grange's to Liapunov's are of little help. the moving frame, not in the lab. sate this effect on a negative-trail bike. But I claim to have made substantial How broad is also a matter of personal progress on the equilibrium theory of the * The first reference (Am. J. of Physics) taste. Short-trail bikes are easy to ride bike, inasmuch as it is the first time we rejected my paper because, wrote the re- but, for straight-line races, people prefer have realistic potential curves showing viewer, "[I] doubt that a bicycle pos- long so that they can forget the the equilibrium riding conditions and sesses an inherent stability." Now, with steering and concentrate on speed. from which one can infer (my Figure 5) Jim, I have come all the way around: he qualitative information concerning the ef- argues the bicycle is intrinsically * To my knowledge, bikes have little fect of front fork geometry on the handle- stable. I doubt anyone can make a frictional torque except for the front- bar torque. [short] article semantically fool-proof wheel ground contact compounded by Jim asks for further calculations. and I [plead] for a minimum of fairness side effects of the trail. Were it not for Thanks for his confidence. I may think from the reader. When Jim argues that this, on a zero-trail bike, the derivative about it sometime, if, as it stands, this an unmoving bicycle 'supported against of my potential curve with respect to a paper is deemed worthy of publication. tilting' is stable, there is no doubt would indeed give the handlebar torque. However, the full theory may not be as about it. Now, what will happen if it is But it is not so easy to take this friction clear and accessible [as] Jim would wish not supported anymore? into account and, moreover, on a posi- it to be, even though it were limited to Static or moving, the area of the poly- tive-trail bike, one would have to take small angle variations. gon of support of a bike is zero (line into account the straightening effect of You were looking for a common ac- between front and rear-wheel ground the trail. This is the reason I cannot sim- quaintance. I have found one. Charles contacts). Notwithstanding the gyro- ply identify the derivative with the han- Hyde-Wright works with us here on the scopic forces Jim refers to, equilibrium dlebar torque, although it is roughly so. linac. He read with interest both my arti- has to be learned. This is what I mean cle and Jim's criticisms and concluded that by intrinsically unstable. I agree that I have not read Sharp's calculations but we were essentially in agreement except gyroscopic forces play a role (in no- I would appreciate a copy. for some minor points of semantics - hands riding); so does the shift of the namely stability/equilibrium. He made line of support left or right when the In summary, I do not claim the several suggestions and, with Jim's criti- front wheel turns (in static, standing problem is solved in full generality. Re- cism, I hope this new version will now equilibrium), but they are minor com- member, the bike is a doubly non- be more palatable, if not fool-proof. pared to the centrifugal force. It is diffi- holonomic system with five degrees of cult to ride hands-off when the front freedom (three only in infinitesimal mo- [Editor'snote: Charles Hyde-Wright is a wheel is heavily loaded and inertia and tion, one of them being the rotation of former president of the Boston Area Bicy- the Coulomb (friction) force between the rear wheel in its instantaneous plane, cle Coalition and was recently graduated ground the front tire overload the gyro- which is of course not taken into ac- from MIT with a PhD.] scopic effect. Similarly, the second ef- I fect will be absent on a zero-trail bike Announcing. . . which is nonetheless perfectly rideable, The 13th Annual if not so leisurely, hands off. But I will not argue with Jim about a International Human Powered point on which we agree: under proper Speed Championships dynamical conditions, a moving bike September 23 - 27, 1987 (faster than, say, about 1 m/sec for a Washington, D.C. riderless bike) with or without rider is 200 Meter Sprints * Practical Vehicle Competition * Water- stable according to my potential curves craft Events · Aircraft Demonstrations * Road Races * All- and [the wording of] Lagrange's theo- Terrain Vehicle Display * "Mini" Symposium * Election of rem. Board Members · Annual Membership Meeting for registration packet write: · Should the potential maximum be broad? Neither ... Now tell me, how IHPVA broad is broad? The zero-trail bike has P.O. Box 51255 what I call a broad maximum. Yet, it is Indianapolis, Indiana 46251-0255 USA not called stable because it is difficult to ride hands-off (any bump on the road L - nage 20 HUMAN PnWER. Srin. 19837 vol. no. ·- e 2U POE... - 17 vo..6 1...... D.I.Y. The construction of the frame is illustrated. We leave the design of additions to you. The simplest is a sheet of plywood rivetted to the frame, with loads secured by a web of rubber cords. The frame can be constructed of either aluminum, in which case the trailer will weigh about 8 kg., or steel, weighing about N 11 kg. The aluminum construction is shown here.

Materials required include:

· Square aluminum tube, 30x30x2mm - between 1"and 1.25" square, between 1/16 and 3/32" thick - about 6m total length (under 20') * Aluminum sheet, 250x1000x2mm - say 1' x 3' x 3/32" * about 350 blind ("Pop") rivets 'type eJrs .n.o.l 4x8 with stainless-steel shafts" (e.g. 3/16" x 3/8" long) * two bicycle wheels of similar diameter to that of the bike I a small ball-joint coupling, e.g. Frame of the Flunder Trailer 20mm (3/4 - 1") diameter; Construction of the Flunder Bike Trailer * a nylon or similar strap (see As mentioned briefly in HP 5/3, Falk had a regular "push-"handle with an illustration) to hold the trailer if a fall Riess and Rainer Pivit of the Bicycle Re- attachment behind the saddle, insteadof disengages the ball joint; search Group, University of Oldenburg, the clever one-sided balljoint near the * four bolts, nuts and washers, about Ilest Germany, developed a rear wheel center shown here. Dave Wil- 6mm (1/4") diameter and 25mm (1") that was shown on the North German TV son. long to bolt the socket part of the show, "Market of Ideas." There were many coupling to the bike frame; requestsfor theirplans, which were dis- CAUTIONS AND DISCLAIMERS * glue, e.g. epoxy or acrylic, to use tributedat no cost by the TV network. This design was produced by thor- between the joints before rivetting. Falk and Rainer gave me a set in Van- ough and conscientious effort by the Bi- couver; we showed an extract in the last cycle Research Group, but the group can- [Note: the ball and socket HP, requesting help in translation. not be held responsible for any failures in coupling is apparentlyobtainable in Among the several who were kind enough design or performance. It is for individu- German bike and motor-cycle stores. to offer was Otto Brodtrick, a recumbent al, not commercial use. Bicycle wheels If you can't get something similar, enthusiast working for the government in used in the trailer are not designed to ingenuity will surely fill the void. I Ottawa, Canada,and presently spending a handle side loads so that sharp turns or use tennis ballsfor duties like this. year at Harvard.He dictated a translation bumps or riding along a slope should not Ed.] to a tape recorder. I am giving a highly ed- be attempted with a loaded trailer. I ited version here, with comments. I am a (Dave) will add my own caution that bi- All joints are stiffened by long-term enthusiast of bicycle trailersfor cycle brakes are often marginal, and to aluminum plate, glued and rivetted as carrying heavy loadsfor short distances, add a heavy, unbraked trailer that can ac- shown on the sketch. The designers making my first when I was 13 to supply centuate the tendency to lift the rear set the wheel-spindle dropouts with our WWII victory gardens with water and wheel off the ground in heavy braking is the openings upwards, partly to lower malodorous varieties offertilizer. When I greatly to reduce one's factor of safety. the center of gravity, and partly to returnedfrom two years in Nigeria, I de- But used with a great deal of extra care, a ensure that the users had the wheels signed a bike trailerfor the organization trailer can enable one to move loads on tight. I (Dave) would feel like VITA for use in developing countries. It much greater (e.g. over 50 kg.) than is adding a retainer in case the once-tight had much in common with Flunder, but possible with a regular bicycle alone. spindle became loose.